Discuss Aristotle's doctrine of the mean

The mean is the result of moral virtues being balanced within the individual. Aristotle saw the mean as the middle road to happiness. He argued that all of life is really an attempt to find the highest good. Pleasure is momentary, but happiness is an ethical state of balance of the individual soul.

Explain the role Aristotle assigns individuals for removing their own ignorance

Although he felt teaching was necessary to achieve this goal. Aristotle placed a strong responsibility upon the head of the individual for removing their own ignorance. He stressed that happiness was the utmost moral goal of every individual, and striving for such a balanced and virtuous state was the unique characteristic that set humanity apart from the beasts (and slaves and women, in his view). Thus, human beings had a responsibility to achieve knowledge, one of the components of true happiness.

What is Aristotle's definition of friendship, and what are the kinds (of friendship) he discusses?

Friendship is the result of and necessary to lead a virtuous life. Rather than friendships achieved for social utility or pure pleasure, the best friendships are moral, and formed out of pure like-mindedness.

Kant frames the categorical imperative in different...

Do the different statements of the supreme principle of ethics cohere well together or are they contradictory?
Kant's most famous definition of the imperative is that nothing can possibly be conceived in the world or out of it that can be called good without qualification except a good will, in other words, intention is what matters, not the result of the act. Kant argues for interiority, and against a focus on practical results over and over throughout his text in ways that ultimately, but controversially cohere.

Argue, using only the categorical imperative, for this conclusion: It is wrong to cheat on this test.

It is wrong to cheat on this test, because to do so would come from an ill will, even if it gained the recipient an a, it would do nothing to prove the purpose of the test, namely the state of the student's knowledge, and it would violate the bond of trust between student and teacher

What does Kant say for helping those who need our help (when we are in a position to help)?

It is imperative to help those in need when we are in a position to do so, regardless of whether this is legally incumbent upon us, it is morally incumbent upon us. One must act as though every action was a universal law, as one would like one's self to be aided when in need, by others.